Realizations of Lattice Quotients of Petrie-Coxeter Polyhedra*

Home , Schlegel diagram

ISSN 2590-9770 The Art of Discrete and Applied Mathematics 4 (2021) #P3.04 https://doi.org/10.26493/2590-9770.1358.428 (Also available at http://adam-journal.eu)

Realizations of lattice quotients of Petrie-Coxeter polyhedra*

Gabor´ Gevay´ † Bolyai Institute, University of Szeged, Aradi vertan´ uk´ tere 1, H-6720 Szeged, Hungary

Egon Schulte‡ Department of Mathematics, Northeastern University, Boston, MA 02115, USA

Received 19 February 2020, accepted 01 June 2020, published online 23 August 2021

Abstract The Petrie-Coxeter polyhedra naturally give rise to several inﬁnite families of ﬁnite regular maps on closed surfaces embedded into the 3-torus. For the dual pair of Petrie- Coxeter polyhedra {4, 6 | 4} and {6, 4 | 4}, we describe highly-symmetric embeddings of these maps as geometric, combinatorially regular polyhedra (polyhedral 2-manifolds) with convex faces in Euclidean spaces of dimensions 5 and 6. In each case the geometric symmetry group is a subgroup of index 1 or 2 in the combinatorial automorphism group.

IN MEMORY OF BRANKO GRUNBAUM¨ .

Keywords: Regular polyhedron, regular map, Petrie-Coxeter polyhedron, polyhedral embedding, polyhedral 2-manifold, automorphism group. Math. Subj. Class.: 51M20, 52B70.

*The authors are grateful to Marston Conder for performing computations on the automorphism groups of the regular maps studied in this paper, for genus up to 1001. The computations have provided valuable insights into the structure of these groups. We would also like to thank the referees for their careful reading of the manuscript and for their useful comments that have helped improve the paper. †Corresponding author. Supported by the Hungarian National Research, Development and Innovation Ofﬁce, OTKA grant No. SNN 132625. ‡Supported by the Simons Foundation Award No. 420718. E-mail addresses: [email protected] (Gabor´ Gevay),´ [email protected] (Egon Schulte) cb This work is licensed under https://creativecommons.org/licenses/by/4.0/ 2 Art Discrete Appl. Math. 4 (2021) #P3.04

1 Introduction In the 1930’s, Coxeter and Petrie discovered three remarkable infinite regular polyhedra (apeirohedra) with convex faces and skew vertex-figures in Euclidean 3-space E3 (see [7]). These polyhedra are often called the Petrie-Coxeter polyhedra and are known to be the only infinite regular polyhedra in E3 that have convex faces (see [10, 11, 16, 23]). The Petrie-Coxeter polyhedra naturally give rise to several infinite families of finite regular maps (abstract regular polyhedra) on closed surfaces embedded into the 3-torus. In this paper we focus on the dual pair of Petrie-Coxeter polyhedra {4, 6 | 4} and {6, 4 | 4} (see Section3 for notation) and investigate the maps in certain infinite families in more detail. These maps occur in dual pairs of types {4, 6} and {6, 4}, and have genus 1 + 2t3 3 and automorphism groups Dt o D6, t ≥ 1. In particular, we derive presentations for their automorphism groups and prove isomorphism with Coxeter’s regular maps {4, 6 | 4, 2t} and their duals [7, p. 57], which were rediscovered in [23, pp. 259]. For t ≥ 3, we describe highly-symmetric polyhedral realizations as geometric polyhedra with convex faces in dimensions 5 and 6. These geometric polyhedra are free of self-intersections and are polyhedral 2-manifolds (polyhedral embeddings) in the sense of [5, 25, 26]. The 6-dimensional polyhedra are particularly remarkable, in that these are combinatorially regular polyhedra whose geometric symmetry group is a subgroup of index 1 or 2 in the full automorphism group. For each map of type {4, 6} or {6, 4}, exactly one of the 6-dimensional realizations is a geometrically regular polyhedron (in this case the index is 1). In all other cases, the realizations are “combinatorially regular polyhedra of index 2” (that is, combinatorially regular polyhedra with the property that exactly one half of all combinatorial symmetries is realized by geometric isometries of the ambient space [9, 19, 37]). The study of polyhedral 2-manifolds has attracted a lot of attention. Remarkable and beautiful 3-dimensional polyhedral embeddings of regular maps of relatively small genus have been discovered (see [33] for a survey of the maps up to genus 6, as well as [32] and [3]), including realizations for some well-known regular maps such as Klein’s maps {3, 7}8 [30] and {7, 3}8 [22] of genus 3 (the latter realized with non-convex planar faces), Dyck’s maps {3, 8}6 of genus 3 [4,1], Coxeter’s maps {4, 6|3} and {6, 4|3} of genus 6 [31], and quite recently, Hurwitz’s regular map of type {3, 7} and genus 7 (sometimes called the Macbeath map) [2], which is the second smallest among the Hurwitz maps (regular maps of type {3, 7}), the smallest being Klein’s {3, 7}8. These realizations naturally have small symmetry groups compared with the much larger automorphism group. Appealing topological pictures of regular maps have been created in [35, 36] and [34]. There are also interesting infinite families of regular maps that have polyhedral embeddings in E3. In [25, 26], two remarkable infinite series of polyhedra of types {4, r} and {r, 4}, r ≥ 3, and genus 1 + (r − 4)2r−3 were described, and these are polyhedral embeddings of Coxeter’s regular maps {4, r | 4br/2c−1} and their duals [24]. The maps in each series also admit polyhedral embeddings in E4 as subcomplexes of the boundary complex of certain convex 4-polytopes [20, 21, 29].

2 Maps, polyhedra, and polyhedral realizations A map M on a closed surface S is a decomposition (tessellation) of S into non-overlapping simply-connected regions, called the faces of M, by arcs, called the edges of M, joining pairs of points, called the vertices of M, such that two conditions are satisfied: first, each edge belongs to exactly two faces; and second, if two distinct edges intersect, they meet in G. Gevay´ and E. Schulte: Realizations of lattice quotients of Petrie-Coxeter polyhedra 3 one vertex or in two vertices (see [15,8]). For most maps on surfaces, the underlying set of vertices, edges, and faces, ordered by inclusion, forms an abstract polyhedron (abstract polytope of rank 3). As we will explain in a moment, conversely, every (locally finite) abstract polyhedron gives rise to a map on a surface. Following [23] (with minor modifications), an abstract polyhedron P is a partially ordered set with a rank function with range {0, 1, 2}. For j = 0, 1, 2, an element of P of rank j is called a j-face of P, or vertex, edge, and face, respectively. A flag of P is an incident triple consisting of a vertex, an edge, and a face. Two flags are adjacent if they differ by one element. Further, P is strongly flag-connected, meaning that any two flags Φ and Ψ of P can be joined by a sequence of flags Φ = Φ0, Φ1,..., Φk = Ψ, where the flags Φi−1 and Φi are adjacent for each i ≥ 1, and Φ ∩ Ψ ⊆ Φi for each i ≥ 0. Finally, every edge contains exactly two vertices and lies in exactly two faces, and every vertex of every face lies in exactly two edges of that face. From now on, we are making the implicit assumption that all maps considered in this paper are abstract polyhedra. For a discussion about this polytopality assumption see also [13]. Every (locally finite) abstract polyhedron lives naturally as a map on a closed surface. This surface is the underlying topological space of the order complex (“combinatorial barycentric subdivision”) of P. Recall that the order complex is the 2-dimensional abstract simplicial complex, whose vertices are the vertices, edges and faces of P, and whose simplices are the chains (subsets of flags) of P (see [23, Ch. 2C]). The maximal simplices are in one-to-one correspondence with the flags of P, and are 2-dimensional (triangles). Adjacency of flags in P corresponds to adjacency of triangles on the surface. We require further terminology and notation that applies to both abstract polyhedra and maps. For 0 ≤ j ≤ 2, every flag Φ is adjacent to just one flag, denoted Φj, differing by the j j-face; the flags Φ and Φ are said to be j-adjacent to each other. For integers j1, . . . , jl (with l ≥ 2 and 0 ≤ j1, . . . , jl ≤ 2) we inductively define the new flag

Φj1...jl := (Φj1...jl−1 )jl .

Note that Φ, Φj1 , Φj1j2 ,..., Φj1...jl is a sequence of successively adjacent flags. If each face of an abstract polyhedron P (or a map M) has p vertices and each vertex lies in q faces, then P (or M) is said to be of (Schlafli¨ ) type {p, q}. An abstract polyhedron P (or a map M) is called regular if its automorphism group Γ(P) (or Γ(M)) is transitive on the flags. Suppose P is an abstract regular polyhedron and Φ := {F0,F1,F2} is a (fixed) base flag of P. Then Γ(P) is generated by distinguished generators ρ0, ρ1, ρ2 (with respect to Φ), where ρj is the unique automorphism which fixes j all elements of Φ but the j-face. Thus ρj(Φ) = Φ for each j = 0, 1, 2. These generators satisfy the standard Coxeter-type relations

2 2 2 p q 2 ρ0 = ρ1 = ρ2 = (ρ0ρ1) = (ρ1ρ2) = (ρ0ρ2) = 1 (2.1) determined by the type {p, q}; in general there are also other independent relations. A k-hole of an abstract polyhedron P (or a map M) is an edge-path which leaves a vertex by the k-th edge from which it entered, in the same sense (that is, always keeping to the left, say, in some local orientation). If a regular polyhedron P has k-holes of length hk, then the distinguished generators of Γ(P) satisfy the additional relation

k−1 hk (ρ0ρ1(ρ2ρ1) ) = 1. (2.2) 4 Art Discrete Appl. Math. 4 (2021) #P3.04

When k = 2 we refer to a k-hole simply as a hole. Thus a hole of a regular polyhedron h of length h contributes the relation (ρ0ρ1ρ2ρ1) = 1.A Petrie polygon of P (or M) is an edge-path with the property that any two successive edges, but not three, are edges of a t face of P. A Petrie polygon of length t contributes the relations (ρ0ρ1ρ2) = 1. For the purpose of this paper, a geometric polyhedron P is a closed surface embedded in a Euclidean space made up of ﬁnitely many convex polygons, the faces of P , such that any two distinct polygons intersect, if at all, in a common vertex or a common edge (see [15,5]). Thus P has convex faces and is free of self-intersections. We require additionally that no two faces with a common edge lie in the same 2-dimensional plane. In [5], these polyhedra were called polyhedral 2-manifolds. We usually identify P with the map on the underlying surface, which, in our applications, will be orientable, or with the abstract polyhedron consisting of the vertices and edges (of the polygons) and the faces, partially ordered by inclusion. We also call P a polyhedral realization of the map or abstract polyhedron. By G(P ) we denote the geometric symmetry group consisting of the isometries of the ambient space that map P to itself. A geometric polyhedron P is geometrically regular if G(P ) acts transitively on the ﬂags. The edge-graph of a (geometric or an abstract) polyhedron is also called the 1-skeleton of the polyhedron; its vertices and edges are formed by the vertices and edges of the polyhedron. Similarly, for a convex d-polytope K and 0 ≤ k ≤ d, the k-skeleton of K is the subcomplex of the face-lattice of K consisting of the faces of dimension less than or equal to k. A convex hexagon is called semi-regular if its symmetry group acts transitively on the vertices. In a semi-regular hexagon which is not regular, the edges are of two kinds and alternate in length.

3 The Petrie-Coxeter polyhedra revisited

Among the geometrically regular polyhedra in Euclidean 3-space E3, the three Petrie- Coxeter polyhedra {4, 6 | 4}, {6, 4 | 4} and {6, 6 | 3} are characterized as the infinite polyhedra (apeirohedra) with convex faces (see [7, 23]). Each polyhedron {p, q | h} forms a periodic polyhedral surface which bounds a pair of congruent non-compact “polyhedral handlebodies” (the inside and outside) which tile E3. Each polyhedron has p-gonal convex faces and q-gonal skew vertex-figures, and has 2-holes of length h. The polyhedra were discovered by Petrie and Coxeter in 1930’s (see [7]). Like any geometrically regular polyhedron in E3, a Petrie-Coxeter polyhedron P has a flag-transitive symmetry group G = G(P ) isomorphic to the combinatorial automorphism group Γ(P ). The translation subgroup T (P ) of G(P ) is generated by three translations in independent directions and can be identified with the 3-dimensional translation lattice Σ = Σ(P ) in E3 spanned by the three corresponding translation vectors. In this paper we focus on the dual pair of polyhedra {4, 6 | 4} and {6, 4 | 4}. The duality is very explicit in this case, in that, up to similarity, either polyhedron could be constructed from the other by choosing the vertices at the face centers of the other; if related in this fashion, the polyhedra share the same symmetry group and translation subgroup. The translation lattice Σ for either polyhedron is the body-centered cubic lattice and thus contains a translation sublattice Λ = Λ(P ) of index 4 generated by three orthogonal vectors of equal lengths. When convenient, we may take Λ to be the standard integral lattice Z3, and Σ to 1 1 1 be the lattice generated by the vectors (1, 0, 0), (0, 1, 0), ( 2 , 2 , 2 ) (denoted Λ(1,1,1) in [23, G. Gevay´ and E. Schulte: Realizations of lattice quotients of Petrie-Coxeter polyhedra 5

Section 6D] and [28]). Note that Λ is also a normal subgroup of G(P ). The polyhedron {6, 4 | 4} is built from copies of an Archimedean truncated octahedron (or rather, parts of its surface). An Archimedean truncated octahedron Q has eight regular hexagons and six squares as faces [12]. Its eight regular hexagons (and their vertices and edges) form a subcomplex of the boundary complex of Q which we will call a hexagonal octet complex, or simply, an HO complex (see Figure1). Clearly, both Q and its HO complex are naturally inscribed in a cube C bounded by the planes containing the six square faces of Q. Then the polyhedron {6, 4 | 4} can be constructed from the standard cubical tessellation T in E3 by translates of C by inscribing copies of the HO complex into the cubical tiles of T (see Figure2). The copies of the HO complex placed into adjacent cubical tiles attach along the “missing square faces” of the corresponding truncated octahedra, and thus a polyhedral surface is formed. Note that the symmetry group of an HO complex coincides with that of its underlying truncated octahedron.

Figure 1: Hexagonal octet complex (HO complex).

Note that the polyhedron {6, 4 | 4} is also related to the Voronoi tiling of the body- centered cubic lattice [7], whose tiles are truncated octahedra tessellating the “outside” and the “inside” of the polyhedron. This tiling is among the 28 uniform tilings of E3 enumerated by Grunbaum¨ [17]. The polyhedron then consists of all the hexagonal 2-faces of this tiling. The dual polyhedron {4, 6 | 4} can also be constructed from a standard cubical tessellation T of E3. In this case, a copy of the 1-skeleton of a smaller cube of half the size is placed concentrically into each cubical tile of T , and then the copies in adjacent cubical tiles of T are connected by cylindrical square-faced tunnels so that, at either end, a tunnel meets the corresponding copy of the 1-skeleton in a “missing square face” (see Figure3). Alternatively, and less formally, each cubical tile of T is shrunk concentrically to half its size and then adjacent shrunk cubes are connected by cylindrical square-faced tunnels. We refer to the part of the polyhedron lying inside a single cube of T as a 6-elbow. Thus a 6- elbow consists of six half-tunnels attached to a shrunk cube in a cross-like fashion, where each half-tunnel is formed by four rectangles with sides of lengths 1 and 1/2 emanating from a square face of the shrunk cube. The construction shows that the polyhedron has a natural 6-elbow decomposition. The polyhedron {4, 6 | 4} also admits a 3-elbow decomposition relative to the standard 6 Art Discrete Appl. Math. 4 (2021) #P3.04

Figure 2: A 3 × 3 × 3 block of the Petrie-Coxeter polyhedron {6, 4 | 4}.

3 3 cubical tessellation T of E . Let e1, e2, e3 denote the standard basis vectors of E , let 1 ai := 2 ei for i = 1, 2, 3, and let

C := {(x1, x2, x3) | 0 ≤ x1, x2, x3 ≤ 1}.

1 3 Then {4, 6 | 4} can be constructed as follows. The vertex set is again 2 Z . The face set of {4, 6 | 4} consists of the translates, under Z3, of a set of twelve particular faces contained in C. More precisely, the intersection of the entire polyhedron with C, denoted E, consists 1 of the twelve square faces of the smaller cubes ai + 2 C (i = 1, 2, 3) which do not lie in 1 1 1 1 the planes xi = 2 or xi = 1. Six of these square faces of E meet at the vertex ( 2 , 2 , 2 ) of the polyhedron. The entire polyhedron then consists of all translates z + E with z ∈ Z3. Each translate z + E coincides with the intersection of the polyhedron with the cubical tile z +C of T . We refer to these intersections as 3-elbows. Then it is clear that the polyhedron admits a natural 3-elbow decomposition such that every cubical tile of T contributes exactly one 3-elbow.

4 Petrie-Coxeter-type polyhedra in the 3-torus The three Petrie-Coxeter polyhedra P naturally give rise to regular maps embedded in the 3-torus which usually are abstract regular polyhedra. This construction was also indepen- dently described by Montero [28]. Suppose as above that the translation subgroup T (P ) of the symmetry group G(P ) of P is identiﬁed with the translation lattice Σ = Σ(P ). Then the maps arise as quotients of P by subgroups of T (P ) which are normal in G(P ), or equivalently, as quotients by sublattices Ω of Σ, denoted P/Ω, which are invariant under G(P ). G. Gevay´ and E. Schulte: Realizations of lattice quotients of Petrie-Coxeter polyhedra 7

Figure 3: The Petrie-Coxeter polyhedron {4, 6 | 4}.

In this paper, we only consider the maps derived from the dual pair of polyhedra {6, 4 | 4} and {4, 6 | 4}, and focus on those that later are realized polyhedrally. We choose, as sublattices Ω, certain scaled copies of the sublattice Λ of Σ (of index 4) described earlier, where here we identify Λ with Z3. The resulting maps are embedded in the 3-torus E3/Ω determined by Ω. We have not investigated quotients of P by other types of sublattices Ω of Σ.

3 4.1 The polyhedron Pt := {6, 4 | 4}/tZ We begin with the polyhedron P := {6, 4 | 4}. Suppose P has been constructed from a standard cubical tessellation T by inscribing HO complexes into the cubical tiles of T so that the HO complexes in adjacent cubes are attached along “missing squares”. Further, suppose that the origin o lies at the center of one of these HO complexes, at HOo (say), and 3 that Λ is generated by the translations by e1, e2, e3, the standard basis vectors of E . Thus Λ = Z3. 3 Now let t ≥ 1 be an integer, and let Λt := tΛ = tZ . Then Λt is invariant under G(P ) and hence Pt := P/Λt = {6, 4 | 4}/Λt 3 is a regular map of type {6, 4} embedded in the 3-torus E /Λt. The numbers of vertices, 3 3 3 edges and faces of Pt are given by 12t , 24t and 8t , respectively. Thus the Euler characteristic is 3 3 3 3 χ(Pt) := 12t − 24t + 8t = −4t , 3 and since Pt is orientable (as we will see), it must have genus 1 + 2t . Clearly, since Pt is 8 Art Discrete Appl. Math. 4 (2021) #P3.04

a regular map with 4-valent vertices, the order of the automorphism group Γ(Pt) of Pt is given by 96t3 = 8 · 12t3. The map is an abstract polyhedron for each t ≥ 1. When t = 1 the map has the property that any two hexagon faces that meet in an edge also meet in the opposite edge, so in particular the map cannot admit a polyhedral realization with convex faces in any Euclidean space. The structure of Γ(Pt) can be determined more explicitly. First note that ∼ ∼ Γ(Pt) = Γ(P/Λt) = Γ(P )/Λt = G(P )/Λt; (4.1) the second equality follows, for example, from [23, 2E18]. Clearly, by construction, Γ(Pt) 3 contains a normal abelian subgroup Zt , where Zt := Z/tZ. This subgroup is generated by the three “translations” corresponding to three generators of Λ/Λt. Its normality is inher- ited from the normality of Λt in G(P ). It follows directly from (4.1) that a presentation for Γ(Pt) can be obtained from a presentation of G(P ) = Γ(P ) by adding a single extra relation determining the quotient of P by Λt. More explicitly, since Λt is generated by the conjugates of the translation by the single vector te1 (or by te2, or by te3), denoted T (say), it suffices to express this translation in terms of the standard generators of G(P ) and add the corresponding relation to the standard relations for G(P ). The details are as follows. We begin by choosing a base flag Φ := {F0,F1,F2} of P = {6, 4 | 4}. Then G(P ) 3 is generated by three involutory isometries R0, R1, R2 of E defined by the conditions Rj(Fi) = Fi, Ri(Fi) 6= Fi, for i, j = 0, 1, 2, j 6= i. The generators R0 and R2 are plane reflections and R1 is a half-turn. In terms of these generators, G(P ) has the presentation

2 2 2 6 4 2 4 R0 = R1 = R2 = (R0R1) = (R1R2) = (R0R2) = (R0R1R2R1) = 1. (4.2)

Note that the relator R0R1R2R1 of the last relation shifts a 2-hole of P one step along itself and hence has period 4. To explain the additional relation for the automorphism group Γ(Pt) of the quotient Pt we use the notation for flags introduced in Section2. Consider the flags Φ and Φ101012101012 of the original polyhedron P , as well as the corresponding implied sequence of successively adjacent flags of P joining them, Φ, Φ1, Φ10, Φ101, Φ1010,..., Φ10101210101, Φ101012101012.

On the underlying surface of P , flags correspond to triangles of the barycentric subdivision, and adjacency of flags corresponds to adjacency of triangles. Thus the sequence of flags translates into a sequence of triangles that starts with the triangle for the base flag, and inspection shows that the last triangle in the sequence is just the translate of the first triangle under the translation T . The expression of T in terms of the generators R0,R1,R2 then can be derived from the flag sequence. In fact, taking the sequence of indices in reverse order we see that

2 2 T = R2R1R0R1R0R1R2R1R0R1R0R1 = (R2R1(R0R1) )

t 2 2t and hence T = (R2R1(R0R1) ) . Thus, a presentation for Γ(Pt) consists of the relations in (4.2) and the single extra relation

2 2t (R2R1(R0R1) ) = 1. (4.3) For a justiﬁcation of the method employed see [23, Sect. 2B] or [27]. (The proper setting for the above argument is the monodromy group (connection group) of P , but since P G. Gevay´ and E. Schulte: Realizations of lattice quotients of Petrie-Coxeter polyhedra 9 regular, this group is isomorphic to the automorphism group of P .) Note that each deﬁning relation of Γ(Pt) involves an even number of generators. In particular this shows that Pt is orientable. In Section 4.2 we will show that Pt is isomorphic to the orientable regular map {4, 6 | 4, 2t}∗, the dual of Coxeter’s regular map {4, 6 | 4, 2t}, with automorphism group 3 Dt o D6. The length of the Petrie polygons of Pt is given by the order of R0R1R2 in Γ(Pt), k which is 6t. In fact, the element (R0R1R2) of G(P ) lies in Λ if and only if 6 | k, and k hence (R0R1R2) lies in Λt if and only if 6t | k. This can either be seen directly geometrically, or by using a coordinate representation for G(P ) and its generators, for example the presentation of [23, p. 231]. Thus the element R0R1R2 of Γ(Pt) has order 6t. 3 3 Recall from [23, Sect. 6D] that the 3-torus E /Λt admits a regular tessellation by t 3 cubes obtained as the quotient of the standard cubical tessellation {4, 3, 4} of E by Λt; this quotient, {4, 3, 4}/Λt, is called a cubic toroid and is denoted {4, 3, 4}(t,0,0). More informally, {4, 3, 4}(t,0,0) is obtained from a t×t×t block of cubes by identifying opposite sides of the block. Note that we can think of Pt as being constructed from {4, 3, 4}(t,0,0) by inscribing copies of the HO complex into the cubical tiles, in the same way in which copies of the HO complex were inscribed into the cubical tiles of the cubical tessellation of E3 to construct P .

3 4.2 The polyhedron Qt := {4, 6 | 4}/tZ For the dual polyhedron Q := {4, 6 | 4} of P := {6, 4 | 4}, we may assume that its vertices lie at the face centers of P . Then G(Q) = G(P ) and Λ = Λ(Q) = Λ(P ). The distinguished generators of G(Q) are just the distinguished generators of G(P ) taken in reverse order. If we label these new generators of G(Q) by R0,R1,R2 so that Rj(Fi) = Fi, Ri(Fi) 6= Fi, for i, j = 0, 1, 2, j 6= i, and some base ﬂag {F0,F1,F2} of Q, then G(Q) admits the presentation

2 2 2 4 6 2 4 R0 = R1 = R2 = (R0R1) = (R1R2) = (R0R2) = (R0R1R2R1) = 1, (4.4) where the order of the relator R0R1R2R1 gives the length of the 2-hole of Q, which again is 4. 3 If we set again Λt := tΛ = tZ for t ≥ 1, then

Qt := Q/Λt = {4, 6 | 4}/Λt

3 3 is a regular map of type {4, 6} and genus 1 + 2t embedded in the 3-torus E /Λt, and is the dual of Pt := P/Λt. The numbers of vertices, edges, and faces of Qt are given by 3 3 3 8t , 24t , and 12t , respectively. The automorphism groups of Qt and Pt are isomorphic. In particular, if we reverse the order of the generators in (4.2) and (4.3), we ﬁnd that a presentation of Γ(Qt) is given by the relations in (4.4) and the extra relation

2 2t (R0R1(R2R1) ) = 1. (4.5)

Now recall from Coxeter [7, p. 57] that the set of relations in (4.4) and (4.5) abstractly deﬁne the automorphism group Γ (say) of the regular map {4, 6 | 4, 2t} determined by the lengths 4 and 2t of the 2-holes and 3-holes, respectively. Thus {4, 6 | 4, 2t} is the “universal” regular map with 2-holes and 3-holes of lengths 4 and 2t, respectively, and by 10 Art Discrete Appl. Math. 4 (2021) #P3.04

[7, p. 57] and [23, p. 259] is known to be finite and have an automorphism group isomorphic 3 3 to Dt o D6, of order 96t . As Γ(Qt) satisfies all the defining relations of Γ, it must be a quotient of Γ, and since Γ(Qt) has the same order as Γ, it must coincide with Γ. Thus Γ(Qt) = Γ and Qt = {4, 6 | 4, 2t}, and by duality, ∗ Pt = {4, 6 | 4, 2t} .

For t = 2 we obtain Q2 = {4, 6 | 4, 4}, which also occurs (with r = 6) as the ﬁrst non- toroidal map in Coxeter’s inﬁnite series of regular maps {4, r | 4br/2c−1} mentioned in the Introduction. The Petrie polygons of the regular map Qt are of course all of length 6t, just like those of Pt. Note that we can construct Qt from {4, 3, 4}(t,0,0) by inscribing copies of a 6-elbow into the cubical tiles, in the same way in which Q can be obtained by inscribing copies of a 6-elbow into the cubical tiles of the cubical tessellation of E3.

5 Embeddings in E6

The polyhedral embeddings of the Petrie-Coxeter type polyhedra Pt and Qt, t ≥ 3, in Euclidean 6-space E6 will be constructed from an embedding of the 3-torus as a 3- dimensional cubical subcomplex in the boundary complex of a convex 6-polytope Kt. Then, via Schlegel diagrams [18], polyhedral embeddings can also be constructed in Eu- 5 6 clidean 5-space E . We begin by describing the polytope Kt in E . Our construction requires that t ≥ 3.

5.1 A polyhedral embedding of the cubic toroid √ 5 It is well-known√ that the 3-torus admits a natural embedding into the 5-sphere 3 S (of 6 radius 3) as the submanifold M consisting of all points x := (x1, . . . , x6) of E satisfying the equations 2 2 2 2 2 2 x1 + x2 = x3 + x4 = x5 + x6 = 1. This expresses M as the product S1 × S1 × S1 lying in E6. 6 Let t ≥ 3. We now construct a convex 6-polytope Kt in E with vertices in M and ex- ploit its structure to ﬁnd polyhedral realizations of the regular maps Pt and Qt constructed in the previous section. For j = 0, . . . , t − 1, set aj := 2πj/t and deﬁne the points

uj := (cos aj, sin aj, 0, 0, 0, 0),

vj := (0, 0, cos aj, sin aj, 0, 0),

wj := (0, 0, 0, 0, cos aj, sin aj), all of which are contained in M. Then the convex 6-polytope

Kt := conv{uj + vk + wl | j, k, l = 0, . . . , t − 1} (5.1) 1 2 3 is the cartesian product of the three regular convex t-gons Kt , Kt and Kt with vertex-sets {u0, . . . , ut−1}, {v0, . . . , vt−1} or {w0, . . . , wt−1}, respectively, lying in the 2-dimensional coordinate subspaces containing the ﬁrst two, the third and the fourth, or the last two standard coordinate axes of E6. Thus 1 2 3 Kt = Kt × Kt × Kt . G. Gevay´ and E. Schulte: Realizations of lattice quotients of Petrie-Coxeter polyhedra 11

Note that Kt is a 6-cube when t = 4. Now recall that the nonempty faces of a cartesian product of convex polytopes are just the cartesian products of the nonempty faces of its component polytopes [18]. In particular, the cartesian products of three edges from dif- j 3 ferent component polygons Kt give t 3-dimensional faces of Kt that are 3-cubes. When t 6= 4 there are no other 3-faces of Kt which are 3-cubes; however, when t = 4 the polytope Kt is the 6-cube and thus each of its 3-faces is a 3-cube. In any case, the union of these t3 3-faces obtained as products of three edges from different components forms a subcomplex Ct of the boundary complex of Kt which is homeomorphic to the 3-torus. This follows from the fact that Ct is topologically the product of the three unit circles deter- 1 2 3 mined by the boundary complexes of Kt , Kt and Kt . Thus Ct is a 3-dimensional cubical complex embedded into the boundary complex of Kt and is isomorphic to the regular cubic toroid {4, 3, 4}(t,0,0). ∼ When t 6= 4 the symmetry group of Kt is isomorphic to the wreath product Dt o S3 (= 3 3 Dt o S3), of order 48t . However, when t = 4 the symmetry group of Kt is isomorphic to 6 C2 o S6, of order 2 6!; this group still contains a subgroup isomorphic to D4 o S3, which is of index 15. In any case, for each t ≥ 3, the subgroup Dt o S3 of the symmetry group of Kt leaves the cubical complex Ct invariant. In fact, Dt o S3 is the symmetry group of Ct, and being of the right order, is also the full automorphism group of Ct. It acts ﬂag-transitively on Ct.

3 5.2 A polyhedral embedding of Pt = {6, 4 | 4}/tZ We begin by rescaling the 6-polytope so that the 3-cubes in the cubical complex become unit 3-cubes. Each 3-cube in Ct has edge length 2 sin(π/t), which is the edge length of a 0 regular convex t-gon inscribed in a unit circle. Thus the new convex polytope Kt := rKt, 0 with r := 1/(2 sin(π/t)), as well as the corresponding cubical complex Ct := r Ct, have 0 the property that all faces which are 3-cubes are unit 3-cubes. In particular, Ct is a faithful 0 realization of the regular cubic toroid {4, 3, 4}(t,0,0). In other words, Ct is isomorphic to 0 3 {4, 3, 4}(t,0,0), and the automorphism group of Ct (of order 48t ) is realized by a group of Euclidean isometries, namely Dt o S3. 0 Now the construction of the embedding for Pt is straightforward: in each 3-cube of Ct (or rather its support) we inscribe a basic building block for Pt, that is, a copy of an HO complex. Note that the insertion in each 3-cube is unique. Thus we arrive at a polyhedral 6 2-manifold in E with regular convex hexagons as faces, which is isomorphic to Pt and invariant under Dt o S3. We will show in the next section that the latter group provides only one half of all the geometric symmetries of the polyhedral 2-manifold. In fact, the poly- 6 hedral embedding for Pt is a geometrically regular polyhedron in E (with regular convex hexagons as faces), that is, its symmetry group is isomorphic to the full automorphism 3 group Dt o D6. 6 There is also a host of 6-dimensional polyhedral realizations of Pt in E with convex hexagonal faces which are not regular. To construct these we can proceed in the same way as above, but now from a modiﬁed HO complex. The modiﬁed HO complex can be obtained from the HO complex of Figure1 by shrinking or expanding the “missing squares” in a uniform fashion, so that the resulting octet consists of eight congruent semi- regular convex hexagons (with two kinds of edges). The resulting polyhedral 2-manifold has Dt o S3 as its symmetry group, that is, it is a combinatorially regular polyhedron of index 2 in E6. 0 Via a Schlegel diagram of the 6-polytope Kt we can also produce polyhedral realiza- 12 Art Discrete Appl. Math. 4 (2021) #P3.04

5 tions of Pt in Euclidean 5-space E . The (possibly modiﬁed) HO complexes sitting inside 0 the 3-cubes of Ct project to distorted HO complexes sitting inside the 3-dimensional paral- 0 lelepipeds which are the projections of the 3-cubes and make up the image of Ct under the 0 projection. There are no self-intersections, since Ct is a subcomplex of the boundary com- 0 plex of Kt, and Schlegel diagrams faithfully represent the boundary complex of a convex polytope.

3 5.3 A polyhedral embedding of Qt = {4, 6 | 4}/tZ We describe two possible ways of constructing polyhedral realizations of the regular maps 3 Qt = {4, 6 | 4}/tZ derived from Q := {4, 6 | 4}. The first is based on the 3-elbow decomposition of Q, and the second employs the 6-elbow decomposition of Q and has a larger symmetry group. The first construction of the polyhedral embedding for Qt proceeds as follows: in each 0 3-cube of Ct we inscribe a basic building block for Qt, that is, a copy of a 3-elbow. Note that the insertion in each 3-cube is uniquely determined once a 3-elbow has been placed in one 3-cube. Thus we arrive at a polyhedral 2-manifold with square faces in E6 which 0 is isomorphic to Qt and lies in the support of the 3-skeleton of Kt. The symmetry group 3 of this realization is Ct o S3, of order 6t , which is a subgroup of index 16 in the full automorphism group of Qt. Note that the special position of the 3-elbow at a corner inside the unit cube prevents additional symmetries from occuring. The second construction of a polyhedral realization of Qt must proceed in a different way, since adjacent 6-elbows in the 6-elbow decomposition attach along the boundary of half-tunnels, not along 4-cycles of edges as in the previous case. In this case we will 0 construct a realization that lies in the support of the 4-skeleton of the polytope Kt but 0 not in the support of the 3-skeleton. However, the 3-dimensional cubical complex Ct still provides the blueprint for the construction. 0 0 The basic idea is simple but the details are tedious. Let Kt and Ct be as above. We 0 shrink all 3-cubes of Ct concentrically (with respect to their centers, by a fixed scaling 0 factor λ < 1), and then connect the shrunk copies of any two adjacent 3-cubes of Ct by small cylindrical tunnels whose “bases” and “tops” are given by the shrunk copies of the square face (with sides of length λ) shared by the adjacent 3-cubes, and whose side faces are rectangles. The set of all side faces of cylindrical tunnels obtained in this way, as well as their vertices and edges, forms a polyhedral realization of the regular map Qt with rectangular faces (with one side of length λ). Moreover, as we explain below, if the 3-cubes are shrunk to the right size, then the rectangular faces become squares. 0 The realization of Qt lies inside the support of the 4-skeleton of Kt, and we show that 0 it is free of self-intersections. Recall that each 3-cube in Ct is the cartesian product of three 0 0 edges from different component polygons of Kt. Two 3-cubes C1 and C2 of Ct are adjacent if their sets of three edges have two edges in common, I1 and I2 (say), and the remaining two edges, I3 and I4 respectively (say), are adjacent edges of a component polygon, F 0 (say), of Kt meeting at a vertex z (say) of F . In the shrinking process, the common square face I1 × I2 × {z} of C1 and C2 yields a pair of parallel squares of smaller size contained in C1 and C2 (the base and top of the cylindrical tunnel). The cylindrical tunnel connecting 0 these smaller squares lies entirely in the cartesian product F := I1 × I2 × F , which is a 0 0 4-dimensional face of Kt that has C1 and C2 as a pair of adjacent facets. Note that F is a double prism over the regular convex t-gon F . More explicitly, since C1 = I1 ×I2 ×I3 and 0 0 0 0 0 0 0 0 C2 = I1 ×I2 ×I4, their shrunk copies are given by C1 = I1 ×I2 ×I3 and C2 = I1 ×I2 ×I4, G. Gevay´ and E. Schulte: Realizations of lattice quotients of Petrie-Coxeter polyhedra 13

0 respectively, where Ij =: [aj, bj] ⊂ Ij for j = 1, 2, 3, 4 and the labeling is such that the points a3 and a4 are closer to z than b3 and b4; here [aj, bj] denotes the line segment with endpoints aj and bj. It now follows that the four rectangular side faces of the cylindrical 0 0 tunnel connecting C1 and C2 are given by 0 0 {a1} × I2 × [a3, a4],I1 × {a2} × [a3, a4], 0 0 {b1} × I2 × [a3, a4],I1 × {b2} × [a3, a4]. Then it is clear that two cylindrical tunnels inserted during the process could only intersect nontrivially (that is, other than in vertices, or in edges lying in their bases or tops) if 0 0 they lied in the same 4-face F of Kt. Thus the question whether or not there are unwanted intersections reduces to a 4-dimensional problem. 0 0 Each 4-face F = I1 × I2 × F of Kt contains exactly t cyclindrical tunnels, one for each pair of adjacent edges of F . As the shrinking process is uniform, the subgroup Dt o S3 0 of the symmetry group of Kt preserves the realization of Qt. In particular, the symmetry group of F , which is isomorphic to Dt, appears as a subgroup of Dt o S3 and permutes the t cylindrical tunnels in F 0 according to a standard dihedral action. Thus the collection of t 0 cylindrical tunnels inside F is invariant under the dihedral group Dt determined by F . To see that nontrivial self-intersections cannot occur, choose vertices u and v of I1 and 0 I2, respectively, and project F orthogonally along the direction of I1 × I2 onto its 2-face 0 0 {u} × {v} × F . A square face shared by a pair of adjacent 3-cubes from Ct that lie in F is necessarily of the form I1 × I2 × {w}, where w is a vertex of F , and is parallel to its two shrunk copies (the base and top of the cylindrical tunnel) in the 3-cubes. Hence each cylindrical tunnel in F 0 projects orthogonally to a line segment strictly inside {u}×{v}×F . For the cylindrical tunnel associated with C1 and C2 as described above, the line segment is [a3, a4]. There are no nontrivial intersections among the t resulting line segments. Since these line segments were obtained by orthogonal projections of cylindrical tunnels, there also cannot be any nontrivial intersections among the cylindrical tunnels. Hence there are 0 no intersections of cylindrical tunnels inside a 4-face F . Thus the realizations of Qt are free of self-intersections, for all scaling factors λ. As will become clearer in a moment, the above analysis also shows that each realizations for Qt is a subcomplex of the 3-skeleton of a convex 6-polytope (depending on λ) deﬁned as the cartesian product of three semi-regular convex 2t-gon (with two kinds of edges); moreover, the realization of Qt has the same vertex-set as the 6-polytope. The tunnels then are formed by four rectangular faces of 3-dimensional faces (rectangular boxes) given by the cartesian product of three edges of these 2t-gons, with one edge from each 2t-gon and exactly two edges of the same length. This observation provides an alternative proof of the fact that the realization for Qt is free of self-intersections. As in the previous section, in all but one case only one half of the combinatorial symmetries appear as symmetries of the realization. The symmetry group of the realization for Qt is Dt o S3 (even if t = 4), except when the rectangular faces become squares (in which 3 case it is Dt o S3), as can be seen as follows. If the unit 3-cubes are shrunk to smaller 3-cubes of the right size, then all rectangular faces of the realization become squares. In fact, the correct scaling factor λ is the inverse ratio between the edge lengths of a convex regular t-gon and of a convex regular 2t-gon obtained from the t-gon by vertex-truncation. It is straightforward to check that π cos t λ = π . 1 + cos t 14 Art Discrete Appl. Math. 4 (2021) #P3.04

6 Thus Qt admits a polyhedral embedding with square faces in E and then, via a Schlegel diagram, a polyhedral embedding with quadrangular faces in E5. The vertex-set of the 6- dimensional square-faced realization coincides with the vertex-set of the cartesian product of three regular 2t-gons, namely the regular 2t-gons obtained by vertex-truncation from the 0 three regular t-gons appearing as factors in the cartesian product decomposition of Kt. This also shows that this square-faced realization is identical with the realization of Qt outlined in [7, p. 57] and [23, p. 259] and further described below. The square-faced realization of Qt has the desirable property that all combinatorial symmetries are realized by geometric symmetries, or more exactly, that the geometric symmetry group is isomorphic to Γ(Qt) = 3 Dt o D6.

The regular maps Qt = {4, 6 | 4, 2t} are isomorphic to certain abstract polyhedra 2{6},G(s) (see [23, p. 259]). These polyhedra admit 6-dimensional realizations (in the sense of [23, Ch. 5]) as subcomplexes of the 2-skeleton of the cartesian product of three regular 2t-gons (see [23, p. 264] and [7, p. 57]), and thus are free of self-intersections and form a polyhedral 2-manifold. Our approach to realizations of the maps Qt as polyhedral 2-manifolds builds on a 6-dimensional realization of the cubic 4-toroid {4, 3, 4}(t,0,0) and exploits the quotient relationships of Qt and Pt with the Coxeter-Petrie polyhedra {4, 6 | 4} and {6, 4 | 4} in a very explicit manner. In particular, we based our construction of rectangle-faced polyhedral embeddings for Qt on the cartesian product of three regular t-gons, rather than of three regular 2t-gons as in [23]. However, as hinted at before, the rectangular-faced embeddings can also be viewed as subcomplexes of the 3-skeleton of a cartesian product of three semi-regular convex 2t-gons. On the other hand, the square- faced realization of [7, p. 57] and [23, p. 264] has the advantage that all combinatorial symmetries are realized by geometric symmetries.

Note that the 6-dimensional polyhedral embeddings for Qt could also be obtained directly from those of the previous section for their combinatorial duals Pt, by choosing as vertices of an embedding for Qt the centers of the hexagonal faces of the embeddings for Pt and then proceeding according to duality. (This process results in rectangular faces for the realization of Qt. In fact, the four vertices of every face of the realization of Qt must lie on four of the edges of a tetrahedron which has two of its vertices located at the centers of a pair of adjacent 3-cubes, and the other two vertices located at adjacent vertices of the common square face of these 3-cubes. These four edges of the tetrahedron form a skew equilateral 4-gon, and the vertices of the realization for Qt all have the same distance from the center of the 3-cube in which they lie. Hence the vertices are those of a rectangle.) Un- der this dual correspondence, the polyhedral embedding with regular hexagons as faces for Pt corresponds to the square-faced polyhedral embedding for Qt, and in particular, both polyhedral embeddings are geometrically regular polyhedra in E6 with symmetry group 3 Dt o D6. In all other cases, polyhedral embeddings with semi-regular hexagon faces (with two kinds of edges) for Pt correspond to rectangle-faced polyhedral embeddings for Qt, and both kinds give combinatorially regular polyhedra of index 2, with symmetry group Dt o S3.

ORCID iDs

Gabor´ Gevay´ https://orcid.org/0000-0002-5469-5165 Egon Schulte https://orcid.org/0000-0001-9725-3589 G. Gevay´ and E. Schulte: Realizations of lattice quotients of Petrie-Coxeter polyhedra 15

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